Finite Group Databasebeta

← C5×#69705 #14288038

Subgroups — C5×#69705 #14288038

Order 1920.

488 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5515C211yesmaximal, characteristic
2C3×C5×#395C211yesmaximal, characteristic, complemented
3C5×#5228C211yesmaximal, characteristic, complemented
4C5×#5227C211yesmaximal, characteristic
5C5×#4528C211yesmaximal, characteristic, complemented
6C5×#4452C211yesmaximal, characteristic, complemented
7C5×#4451C211yesmaximal, characteristic
8C5×#1603—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×C3×C5×C4⋊C4—21no—
11C2×C3×C5×C4⋊C4C2211yescharacteristic
12C2×C5×#216—21no—
13C2×C5×#215—21no—
14C5×#930C2211yescharacteristic
15C5×#929—21no—
16C4×C5×#67—21no—
17C5×#924C2211yescharacteristic
18C5×#898—21no—
19C5×#896—21no—
20C5×#894—21no—
21C5×#889—21no—
22C3×C5×#95—21no—
23C3×C5×#95C2211yescharacteristic
24C5×#835—21no—
25C5×#835—21no—
26C5×#830C2211yescharacteristic
27C5×#830C2211yescharacteristic
28#69705C511yesmaximal, characteristic, direct product factor, Hall
29C5×#539—31no—
30C5×#451—31no—
31C5×#448—31no—
32C5×#395D311yescharacteristic, complemented
33C5×#328—31no—
34C5×#327—31no—
35C5×#324—31no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—42no—
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5D411yescharacteristic
42C22×C3×C4×C5D411yescharacteristic
43C22×C3×C4×C5C2311yescharacteristic
44C2×C5×#67—21no—
45C2×C5×D12D411yescharacteristic
46C2×C4×C5×D3—21no—
47C2×C5×Dic6D411yescharacteristic
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—42no—
51C5×#217—21no—
52C5×#216—42no—
53C5×#215—42no—
54C5×(C2×D3)⋊C4—21no—
55C5×(C2×D3)⋊C4—21no—
56C5×(C2×D3)⋊C4—21no—
57C5×(C2×D3)⋊C4—21no—
58C5×C4⋊Dic3D411yescharacteristic
59C5×C4⋊Dic3D411yescharacteristic
60C5×Dic3⋊C4—21no—
61C5×Dic3⋊C4—21no—
62C5×Dic3⋊C4—21no—
63C5×Dic3⋊C4—21no—
64C4×C5×Dic3—21no—
65C2×C5×C3⋊C8—21no—
66C2×C5×C3⋊C8—21no—
67#5515C2×C511yescharacteristic
68C3×#395C2×C511yescharacteristic, complemented
69#5228C2×C511yescharacteristic, complemented
70#5227C2×C511yescharacteristic
71#4528C2×C511yescharacteristic, complemented
72#4452C2×C511yescharacteristic, complemented
73#4451C2×C511yescharacteristic
74C22×C5×Q8C2×D311yescharacteristic
75C2×C5×Q16—61no—
76C2×C5×SD16—61no—
77C5×#126—61no—
78C5×#125—61no—
79C5×#123—61no—
80C5×#123—61no—
81C5×C22⋊Q8—31no—
82C5×C22⋊Q8—61no—
83C5×#121—31no—
84C4×C5×D4—61no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4C2×D311yescharacteristic
87C5×#103—61no—
88C5×#103—61no—
89C5×C22⋊C8—31no—
90C5×C22⋊C8—31no—
91C5×#95—21no—
92C5×#95C2×D311yescharacteristic
93#1603—31noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C5×D3—21no—
96C3×C5×Q8—22no—
97C3×C5×Q8—22no—
98C3×C5×Q8—41no—
99C3×C5×Q8—41no—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5—22no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5—41no—
112C2×C3×C4×C5C2×D411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C5×#67—41no—
115C5×#67—41no—
116C2×C5×Dic3—21no—
117C2×C5×Dic3—21no—
118C2×C5×Dic3—21no—
119C5×D12—41no—
120C4×C5×D3—81no—
121C5×Dic6—41no—
122C5×C3⋊C8—41no—
123C5×C3⋊C8—41no—
124C22×C3×Q8C22×C511yescharacteristic
125C2×C3×C4⋊C4—21no—
126C2×C3×C4⋊C4C22×C511yescharacteristic
127C2×#216—21no—
128C2×#215—21no—
129#930C22×C511yescharacteristic
130#929—21no—
131C4×#67—21no—
132#924C22×C511yescharacteristic
133#898—21no—
134#896—21no—
135#894—21no—
136#889—21no—
137C3×#95—21no—
138C3×#95C22×C511yescharacteristic
139#835—21no—
140#835—21no—
141#830C22×C511yescharacteristic
142#830C22×C511yescharacteristic
143C2×C5×Q8—21no—
144C2×C5×Q8—21no—
145C2×C5×Q8—31no—
146C2×C5×Q8—42no—
147C2×C5×D4—31no—
148C2×C5×D4—61no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—61no—
152C22×C4×C5#6711yescharacteristic
153C22×C4×C5#6711yescharacteristic
154C22×C4×C5C22×D311yescharacteristic
155C5×Q16—122no—
156C5×SD16—122no—
157C2×C5×C8—61no—
158C2×C5×C8—61no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—31no—
162C5×C4⋊C4—31no—
163C5×C4⋊C4—42no—
164C5×C4⋊C4—61no—
165C5×C4⋊C4—61no—
166C5×C4⋊C4—61no—
167C5×C4⋊C4—61no—
168C5×C22⋊C4—61no—
169C5×C22⋊C4—61no—
170C5×C22⋊C4—61no—
171C5×C22⋊C4—61no—
172C5×C22⋊C4—61no—
173C42×C5—61no—
174#539—31no—
175#451—31no—
176#448—31no—
177#395C5×D311yescharacteristic, complemented
178#328—31no—
179#327—31no—
180#324—31no—
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5#12011yescharacteristic
184C22×C3×C5#13511yescharacteristic
185C22×C3×C5#13711yescharacteristic
186C2×C5×D3—41no—
187C2×C5×D3—41no—
188C2×C5×D3—41no—
189C3×C4×C5—21no—
190C3×C4×C5—21no—
191C3×C4×C5—42no—
192C3×C4×C5—41no—
193C3×C4×C5—41no—
194C3×C4×C5—42no—
195C5×Dic3—41no—
196C5×Dic3—41no—
197C5×Dic3—41no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—42no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C5×D411yescharacteristic
204C22×C3×C4C5×D411yescharacteristic
205C22×C3×C4C23×C511yescharacteristic
206C2×#67—21no—
207C2×D12C5×D411yescharacteristic
208C2×C4×D3—21no—
209C2×Dic6C5×D411yescharacteristic
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—42no—
213#217—21no—
214#216—42no—
215#215—42no—
216(C2×D3)⋊C4—21no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—21no—
220C4⋊Dic3C5×D411yescharacteristic
221C4⋊Dic3C5×D411yescharacteristic
222Dic3⋊C4—21no—
223Dic3⋊C4—21no—
224Dic3⋊C4—21no—
225Dic3⋊C4—21no—
226C4×Dic3—21no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—21no—
229C23×C5—61no—
230C23×C5C2×#6711yescharacteristic
231C5×Q8—22no—
232C5×Q8—22no—
233C5×Q8—41no—
234C5×Q8—41no—
235C5×Q8—121no—
236C5×D4—121no—
237C5×D4—121no—
238C5×D4—121no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—21no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—41no—
249C2×C4×C5—41no—
250C2×C4×C5—41no—
251C2×C4×C5—61no—
252C2×C4×C5—61no—
253C2×C4×C5—61no—
254C2×C4×C5—241no—
255C2×C4×C5D3×D411yescharacteristic
256C2×C4×C5D3×D411yescharacteristic
257C5×C8—121no—
258C5×C8—121no—
259C22×Q8C2×C5×D311yescharacteristic
260C2×Q16—61no—
261C2×SD16—61no—
262#126—61no—
263#125—61no—
264#123—61no—
265#123—61no—
266C22⋊Q8—31no—
267C22⋊Q8—61no—
268#121—31no—
269C4×D4—61no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C2×C5×D311yescharacteristic
272#103—61no—
273#103—61no—
274C22⋊C8—31no—
275C22⋊C8—31no—
276#95—21no—
277#95C2×C5×D311yescharacteristic
278C2×C3×C5—21no—
279C2×C3×C5—21no—
280C2×C3×C5#45211yescharacteristic
281C2×C3×C5#45611yescharacteristic
282C2×C3×C5B(4,𝔽2)11yescharacteristic
283C5×D3—81no—
284C23×C3C2×C5×D411yescharacteristic
285C22×D3—21no—
286C3×Q8—22no—
287C3×Q8—22no—
288C3×Q8—41no—
289C3×Q8—41no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4C2×C5×D411yescharacteristic
303C2×C3×C4C2×C5×D411yescharacteristic
304#67—41no—
305#67—41no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309D12—41no—
310C4×D3—81no—
311Dic6—41no—
312C3⋊C8—41no—
313C3⋊C8—41no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—121no—
317C22×C5—121no—
318C22×C5—121no—
319C22×C5#93711yescharacteristic
320C22×C5#94211yescharacteristic
321C22×C5#95011yescharacteristic
322C4×C5—21no—
323C4×C5—21no—
324C4×C5—42no—
325C4×C5—42no—
326C4×C5—41no—
327C4×C5—41no—
328C4×C5—121no—
329C4×C5—121no—
330C4×C5—121no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—31no—
334C2×Q8—42no—
335C2×D4—31no—
336C2×D4—61no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—61no—
340C22×C4C5×#6711yescharacteristic
341C22×C4C5×#6711yescharacteristic
342C22×C4C22×C5×D311yescharacteristic
343Q16—122no—
344SD16—122no—
345C2×C8—61no—
346C2×C8—61no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—31no—
350C4⋊C4—31no—
351C4⋊C4—42no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C22⋊C4—61no—
357C22⋊C4—61no—
358C22⋊C4—61no—
359C22⋊C4—61no—
360C22⋊C4—61no—
361C42—61no—
362C3×C5#160311yescharacteristic, complemented, Hall
363C22×C3—21no—
364C22×C3—21no—
365C22×C3C5×#12011yescharacteristic
366C22×C3C5×#13511yescharacteristic
367C22×C3C5×#13711yescharacteristic
368C2×D3—41no—
369C2×D3—41no—
370C2×D3—41no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—42no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—42no—
377Dic3—41no—
378Dic3—41no—
379Dic3—41no—
380C2×C5—21no—
381C2×C5—21no—
382C2×C5—241no—
383C2×C5#503211yescharacteristic
384C2×C5#504711yescharacteristic
385C2×C5#509511yescharacteristic
386C23—61no—
387C23C2×C5×#6711yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—41no—
391Q8—41no—
392Q8—121no—
393D4—121no—
394D4—121no—
395D4—121no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—241no—
412C2×C4C5×D3×D411yescharacteristic
413C2×C4C5×D3×D411yescharacteristic
414C8—121no—
415C8—121no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3C5×#45211yescharacteristic
419C2×C3C5×#45611yescharacteristic
420C2×C3C5×B(4,𝔽2)11yescharacteristic
421D3—81no—
422C5#6970511yescharacteristic, direct product factor, Sylow
423C22—21no—
424C22—21no—
425C22—121no—
426C22—121no—
427C22—121no—
428C22C5×#93711yescharacteristic
429C22C5×#94211yescharacteristic
430C22C5×#95011yescharacteristic
431C4—21no—
432C4—21no—
433C4—42no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—121no—
438C4—121no—
439C4—121no—
440C3C5×#160311yescharacteristic, complemented, Sylow
441C2—21no—
442C2—21no—
443C2—241no—
444C2C5×#503211yescharacteristic
445C2C5×#504711yescharacteristic
446C2C5×#509511yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#395C2
socleC22×C3×C5#120
prime radicalC5×#930C22
hypercenterC5×#395D3
hypocenterC3C5×#1603
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#395

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×C5×#395C2semidirect
3. C5×#5228C2semidirect
5. C5×#4528C2semidirect
6. C5×#4452C2semidirect
28. #69705C5direct (N × H)
32. C5×#395D3semidirect
68. C3×#395C2×C5semidirect
69. #5228C2×C5semidirect
71. #4528C2×C5semidirect
72. #4452C2×C5semidirect
177. #395C5×D3semidirect
362. C3×C5#1603semidirect
422. C5#69705direct (N × H)
440. C3C5×#1603semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×#55152
C3×C5×#3952
C5×#52282
C5×#52272
C5×#45282
C5×#44522
C5×#44512
C5×#16033
#697055